LMFDB - Embedded newform 1449.4.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1449,4,Mod(1,1449)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1449, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1449.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1449 = 3^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1449.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$85.4937675983$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 60x^{7} - 22x^{6} + 1179x^{5} + 694x^{4} - 7936x^{3} - 4352x^{2} + 11008x + 3072 $ x^9 - 60x^7 - 22x^6 + 1179x^5 + 694x^4 - 7936x^3 - 4352x^2 + 11008*x + 3072
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 161)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$5.30303$ of defining polynomial
Character	$\chi$	$=$	1449.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.30303 q^{2} +20.1221 q^{4} -13.2233 q^{5} -7.00000 q^{7} -64.2837 q^{8} +O(q^{10})$	\(q-5.30303 q^{2} +20.1221 q^{4} -13.2233 q^{5} -7.00000 q^{7} -64.2837 q^{8} +70.1237 q^{10} +33.2790 q^{11} -78.9006 q^{13} +37.1212 q^{14} +179.921 q^{16} -61.6777 q^{17} +26.3259 q^{19} -266.081 q^{20} -176.479 q^{22} +23.0000 q^{23} +49.8565 q^{25} +418.412 q^{26} -140.855 q^{28} -171.124 q^{29} +275.295 q^{31} -439.858 q^{32} +327.078 q^{34} +92.5633 q^{35} -381.523 q^{37} -139.607 q^{38} +850.044 q^{40} -344.863 q^{41} -349.834 q^{43} +669.643 q^{44} -121.970 q^{46} -261.666 q^{47} +49.0000 q^{49} -264.390 q^{50} -1587.64 q^{52} +317.144 q^{53} -440.060 q^{55} +449.986 q^{56} +907.476 q^{58} +244.266 q^{59} -29.3338 q^{61} -1459.89 q^{62} +893.207 q^{64} +1043.33 q^{65} -276.252 q^{67} -1241.08 q^{68} -490.866 q^{70} -798.729 q^{71} +329.217 q^{73} +2023.23 q^{74} +529.732 q^{76} -232.953 q^{77} +631.414 q^{79} -2379.16 q^{80} +1828.82 q^{82} -865.620 q^{83} +815.585 q^{85} +1855.18 q^{86} -2139.30 q^{88} +896.337 q^{89} +552.304 q^{91} +462.808 q^{92} +1387.62 q^{94} -348.116 q^{95} +58.9955 q^{97} -259.848 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 48 q^{4} + 4 q^{5} - 63 q^{7} - 66 q^{8}+O(q^{10}) $ 9 * q + 48 * q^4 + 4 * q^5 - 63 * q^7 - 66 * q^8	$ 9 q + 48 q^{4} + 4 q^{5} - 63 q^{7} - 66 q^{8} + 50 q^{10} + 8 q^{11} + 25 q^{13} + 180 q^{16} + 28 q^{17} + 254 q^{19} - 302 q^{20} - 122 q^{22} + 207 q^{23} + 295 q^{25} + 6 q^{26} - 336 q^{28} - 25 q^{29} + 1171 q^{31} - 1018 q^{32} + 8 q^{34} - 28 q^{35} + 70 q^{37} - 282 q^{38} - 54 q^{40} - 221 q^{41} - 42 q^{43} + 1214 q^{44} - 159 q^{47} + 441 q^{49} + 1680 q^{50} - 664 q^{52} + 774 q^{53} + 1498 q^{55} + 462 q^{56} + 842 q^{58} - 1080 q^{59} + 686 q^{61} + 2078 q^{62} - 1260 q^{64} + 1656 q^{65} - 370 q^{67} + 1936 q^{68} - 350 q^{70} - 1035 q^{71} + 1979 q^{73} + 5494 q^{74} + 206 q^{76} - 56 q^{77} + 2336 q^{79} + 242 q^{80} + 1642 q^{82} - 130 q^{83} - 272 q^{85} + 1906 q^{86} - 3742 q^{88} + 1328 q^{89} - 175 q^{91} + 1104 q^{92} + 6078 q^{94} + 484 q^{95} - 104 q^{97}+O(q^{100}) $ 9 * q + 48 * q^4 + 4 * q^5 - 63 * q^7 - 66 * q^8 + 50 * q^10 + 8 * q^11 + 25 * q^13 + 180 * q^16 + 28 * q^17 + 254 * q^19 - 302 * q^20 - 122 * q^22 + 207 * q^23 + 295 * q^25 + 6 * q^26 - 336 * q^28 - 25 * q^29 + 1171 * q^31 - 1018 * q^32 + 8 * q^34 - 28 * q^35 + 70 * q^37 - 282 * q^38 - 54 * q^40 - 221 * q^41 - 42 * q^43 + 1214 * q^44 - 159 * q^47 + 441 * q^49 + 1680 * q^50 - 664 * q^52 + 774 * q^53 + 1498 * q^55 + 462 * q^56 + 842 * q^58 - 1080 * q^59 + 686 * q^61 + 2078 * q^62 - 1260 * q^64 + 1656 * q^65 - 370 * q^67 + 1936 * q^68 - 350 * q^70 - 1035 * q^71 + 1979 * q^73 + 5494 * q^74 + 206 * q^76 - 56 * q^77 + 2336 * q^79 + 242 * q^80 + 1642 * q^82 - 130 * q^83 - 272 * q^85 + 1906 * q^86 - 3742 * q^88 + 1328 * q^89 - 175 * q^91 + 1104 * q^92 + 6078 * q^94 + 484 * q^95 - 104 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−5.30303	−1.87490	−0.937451	−	0.348117i	$-0.886821\pi$
$2$	−5.30303	−1.87490	−0.937451	+	0.348117i	$0.886821\pi$
$3$	0	0
$3$	0	0
$4$	20.1221	2.51526
$5$	−13.2233	−1.18273	−0.591365	−	0.806404i	$-0.701411\pi$
$5$	−13.2233	−1.18273	−0.591365	+	0.806404i	$0.701411\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−64.2837	−2.84096
$9$	0	0
$10$	70.1237	2.21750
$11$	33.2790	0.912182	0.456091	−	0.889933i	$-0.349249\pi$
$11$	33.2790	0.912182	0.456091	+	0.889933i	$0.349249\pi$
$12$	0	0
$13$	−78.9006	−1.68331	−0.841657	−	0.540012i	$-0.818420\pi$
$13$	−78.9006	−1.68331	−0.841657	+	0.540012i	$0.818420\pi$
$14$	37.1212	0.708647
$15$	0	0
$16$	179.921	2.81127
$17$	−61.6777	−0.879943	−0.439972	−	0.898012i	$-0.645012\pi$
$17$	−61.6777	−0.879943	−0.439972	+	0.898012i	$0.645012\pi$
$18$	0	0
$19$	26.3259	0.317872	0.158936	−	0.987289i	$-0.449194\pi$
$19$	26.3259	0.317872	0.158936	+	0.987289i	$0.449194\pi$
$20$	−266.081	−2.97487
$21$	0	0
$22$	−176.479	−1.71025
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	49.8565	0.398852
$26$	418.412	3.15605
$27$	0	0
$28$	−140.855	−0.950679
$29$	−171.124	−1.09576	−0.547879	−	0.836558i	$-0.684565\pi$
$29$	−171.124	−1.09576	−0.547879	+	0.836558i	$0.684565\pi$
$30$	0	0
$31$	275.295	1.59498	0.797490	−	0.603332i	$-0.206161\pi$
$31$	275.295	1.59498	0.797490	+	0.603332i	$0.206161\pi$
$32$	−439.858	−2.42989
$33$	0	0
$34$	327.078	1.64981
$35$	92.5633	0.447030
$36$	0	0
$37$	−381.523	−1.69519	−0.847595	−	0.530644i	$-0.821950\pi$
$37$	−381.523	−1.69519	−0.847595	+	0.530644i	$0.821950\pi$
$38$	−139.607	−0.595980
$39$	0	0
$40$	850.044	3.36010
$41$	−344.863	−1.31362	−0.656812	−	0.754054i	$-0.728096\pi$
$41$	−344.863	−1.31362	−0.656812	+	0.754054i	$0.728096\pi$
$42$	0	0
$43$	−349.834	−1.24068	−0.620339	−	0.784334i	$-0.713005\pi$
$43$	−349.834	−1.24068	−0.620339	+	0.784334i	$0.713005\pi$
$44$	669.643	2.29437
$45$	0	0
$46$	−121.970	−0.390944
$47$	−261.666	−0.812084	−0.406042	−	0.913854i	$-0.633091\pi$
$47$	−261.666	−0.812084	−0.406042	+	0.913854i	$0.633091\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−264.390	−0.747809
$51$	0	0
$52$	−1587.64	−4.23397
$53$	317.144	0.821944	0.410972	−	0.911648i	$-0.365189\pi$
$53$	317.144	0.821944	0.410972	+	0.911648i	$0.365189\pi$
$54$	0	0
$55$	−440.060	−1.07887
$56$	449.986	1.07378
$57$	0	0
$58$	907.476	2.05444
$59$	244.266	0.538994	0.269497	−	0.963001i	$-0.413142\pi$
$59$	244.266	0.538994	0.269497	+	0.963001i	$0.413142\pi$
$60$	0	0
$61$	−29.3338	−0.0615706	−0.0307853	−	0.999526i	$-0.509801\pi$
$61$	−29.3338	−0.0615706	−0.0307853	+	0.999526i	$0.509801\pi$
$62$	−1459.89	−2.99043
$63$	0	0
$64$	893.207	1.74454
$65$	1043.33	1.99091
$66$	0	0
$67$	−276.252	−0.503725	−0.251863	−	0.967763i	$-0.581043\pi$
$67$	−276.252	−0.503725	−0.251863	+	0.967763i	$0.581043\pi$
$68$	−1241.08	−2.21329
$69$	0	0
$70$	−490.866	−0.838138
$71$	−798.729	−1.33509	−0.667547	−	0.744568i	$-0.732656\pi$
$71$	−798.729	−1.33509	−0.667547	+	0.744568i	$0.732656\pi$
$72$	0	0
$73$	329.217	0.527834	0.263917	−	0.964545i	$-0.414986\pi$
$73$	329.217	0.527834	0.263917	+	0.964545i	$0.414986\pi$
$74$	2023.23	3.17831
$75$	0	0
$76$	529.732	0.799531
$77$	−232.953	−0.344772
$78$	0	0
$79$	631.414	0.899236	0.449618	−	0.893221i	$-0.351560\pi$
$79$	631.414	0.899236	0.449618	+	0.893221i	$0.351560\pi$
$80$	−2379.16	−3.32498
$81$	0	0
$82$	1828.82	2.46292
$83$	−865.620	−1.14475	−0.572375	−	0.819992i	$-0.693978\pi$
$83$	−865.620	−1.14475	−0.572375	+	0.819992i	$0.693978\pi$
$84$	0	0
$85$	815.585	1.04074
$86$	1855.18	2.32615
$87$	0	0
$88$	−2139.30	−2.59148
$89$	896.337	1.06755	0.533773	−	0.845628i	$-0.320774\pi$
$89$	896.337	1.06755	0.533773	+	0.845628i	$0.320774\pi$
$90$	0	0
$91$	552.304	0.636233
$92$	462.808	0.524468
$93$	0	0
$94$	1387.62	1.52258
$95$	−348.116	−0.375957
$96$	0	0
$97$	58.9955	0.0617534	0.0308767	−	0.999523i	$-0.490170\pi$
$97$	58.9955	0.0617534	0.0308767	+	0.999523i	$0.490170\pi$
$98$	−259.848	−0.267843
$99$	0	0
$100$	1003.22	1.00322
$101$	360.988	0.355640	0.177820	−	0.984063i	$-0.443095\pi$
$101$	360.988	0.355640	0.177820	+	0.984063i	$0.443095\pi$
$102$	0	0
$103$	−919.874	−0.879979	−0.439990	−	0.898003i	$-0.645018\pi$
$103$	−919.874	−0.879979	−0.439990	+	0.898003i	$0.645018\pi$
$104$	5072.02	4.78224
$105$	0	0
$106$	−1681.82	−1.54106
$107$	−1748.90	−1.58011	−0.790057	−	0.613033i	$-0.789949\pi$
$107$	−1748.90	−1.58011	−0.790057	+	0.613033i	$0.789949\pi$
$108$	0	0
$109$	−1941.95	−1.70647	−0.853235	−	0.521527i	$-0.825363\pi$
$109$	−1941.95	−1.70647	−0.853235	+	0.521527i	$0.825363\pi$
$110$	2333.65	2.02277
$111$	0	0
$112$	−1259.45	−1.06256
$113$	378.205	0.314854	0.157427	−	0.987531i	$-0.449680\pi$
$113$	378.205	0.314854	0.157427	+	0.987531i	$0.449680\pi$
$114$	0	0
$115$	−304.137	−0.246616
$116$	−3443.38	−2.75612
$117$	0	0
$118$	−1295.35	−1.01056
$119$	431.744	0.332587
$120$	0	0
$121$	−223.507	−0.167924
$122$	155.558	0.115439
$123$	0	0
$124$	5539.50	4.01179
$125$	993.647	0.710996
$126$	0	0
$127$	1502.04	1.04948	0.524742	−	0.851262i	$-0.324162\pi$
$127$	1502.04	1.04948	0.524742	+	0.851262i	$0.324162\pi$
$128$	−1217.84	−0.840957
$129$	0	0
$130$	−5532.80	−3.73276
$131$	−1712.14	−1.14191	−0.570955	−	0.820981i	$-0.693427\pi$
$131$	−1712.14	−1.14191	−0.570955	+	0.820981i	$0.693427\pi$
$132$	0	0
$133$	−184.281	−0.120144
$134$	1464.97	0.944436
$135$	0	0
$136$	3964.87	2.49989
$137$	−1467.88	−0.915397	−0.457699	−	0.889107i	$-0.651326\pi$
$137$	−1467.88	−0.915397	−0.457699	+	0.889107i	$0.651326\pi$
$138$	0	0
$139$	−2756.49	−1.68203	−0.841014	−	0.541013i	$-0.818041\pi$
$139$	−2756.49	−1.68203	−0.841014	+	0.541013i	$0.818041\pi$
$140$	1862.57	1.12440
$141$	0	0
$142$	4235.68	2.50317
$143$	−2625.73	−1.53549
$144$	0	0
$145$	2262.83	1.29599
$146$	−1745.84	−0.989637
$147$	0	0
$148$	−7677.04	−4.26384
$149$	−2471.08	−1.35865	−0.679324	−	0.733838i	$-0.737727\pi$
$149$	−2471.08	−1.35865	−0.679324	+	0.733838i	$0.737727\pi$
$150$	0	0
$151$	−1222.97	−0.659101	−0.329550	−	0.944138i	$-0.606897\pi$
$151$	−1222.97	−0.659101	−0.329550	+	0.944138i	$0.606897\pi$
$152$	−1692.33	−0.903064
$153$	0	0
$154$	1235.36	0.646415
$155$	−3640.31	−1.88643
$156$	0	0
$157$	−680.886	−0.346119	−0.173059	−	0.984911i	$-0.555365\pi$
$157$	−680.886	−0.346119	−0.173059	+	0.984911i	$0.555365\pi$
$158$	−3348.40	−1.68598
$159$	0	0
$160$	5816.39	2.87391
$161$	−161.000	−0.0788110
$162$	0	0
$163$	−2235.25	−1.07410	−0.537050	−	0.843550i	$-0.680461\pi$
$163$	−2235.25	−1.07410	−0.537050	+	0.843550i	$0.680461\pi$
$164$	−6939.36	−3.30411
$165$	0	0
$166$	4590.41	2.14629
$167$	457.282	0.211889	0.105945	−	0.994372i	$-0.466213\pi$
$167$	457.282	0.211889	0.105945	+	0.994372i	$0.466213\pi$
$168$	0	0
$169$	4028.30	1.83355
$170$	−4325.07	−1.95128
$171$	0	0
$172$	−7039.38	−3.12063
$173$	−361.973	−0.159077	−0.0795384	−	0.996832i	$-0.525345\pi$
$173$	−361.973	−0.159077	−0.0795384	+	0.996832i	$0.525345\pi$
$174$	0	0
$175$	−348.996	−0.150752
$176$	5987.60	2.56439
$177$	0	0
$178$	−4753.30	−2.00154
$179$	−3215.60	−1.34271	−0.671356	−	0.741135i	$-0.734288\pi$
$179$	−3215.60	−1.34271	−0.671356	+	0.741135i	$0.734288\pi$
$180$	0	0
$181$	2607.72	1.07088	0.535442	−	0.844572i	$-0.320145\pi$
$181$	2607.72	1.07088	0.535442	+	0.844572i	$0.320145\pi$
$182$	−2928.88	−1.19288
$183$	0	0
$184$	−1478.52	−0.592382
$185$	5045.01	2.00495
$186$	0	0
$187$	−2052.57	−0.802669
$188$	−5265.27	−2.04260
$189$	0	0
$190$	1846.07	0.704884
$191$	−100.043	−0.0379000	−0.0189500	−	0.999820i	$-0.506032\pi$
$191$	−100.043	−0.0379000	−0.0189500	+	0.999820i	$0.506032\pi$
$192$	0	0
$193$	1804.25	0.672915	0.336458	−	0.941699i	$-0.390771\pi$
$193$	1804.25	0.672915	0.336458	+	0.941699i	$0.390771\pi$
$194$	−312.855	−0.115782
$195$	0	0
$196$	985.982	0.359323
$197$	5340.92	1.93160	0.965799	−	0.259293i	$-0.0834894\pi$
$197$	5340.92	1.93160	0.965799	+	0.259293i	$0.0834894\pi$
$198$	0	0
$199$	3285.28	1.17029	0.585144	−	0.810929i	$-0.301038\pi$
$199$	3285.28	1.17029	0.585144	+	0.810929i	$0.301038\pi$
$200$	−3204.96	−1.13312
$201$	0	0
$202$	−1914.33	−0.666791
$203$	1197.87	0.414158
$204$	0	0
$205$	4560.24	1.55366
$206$	4878.11	1.64987
$207$	0	0
$208$	−14195.9	−4.73225
$209$	876.100	0.289957
$210$	0	0
$211$	−1893.58	−0.617816	−0.308908	−	0.951092i	$-0.599964\pi$
$211$	−1893.58	−0.617816	−0.308908	+	0.951092i	$0.599964\pi$
$212$	6381.59	2.06740
$213$	0	0
$214$	9274.44	2.96256
$215$	4625.97	1.46739
$216$	0	0
$217$	−1927.06	−0.602846
$218$	10298.2	3.19946
$219$	0	0
$220$	−8854.91	−2.71363
$221$	4866.41	1.48122
$222$	0	0
$223$	−456.269	−0.137014	−0.0685069	−	0.997651i	$-0.521824\pi$
$223$	−456.269	−0.137014	−0.0685069	+	0.997651i	$0.521824\pi$
$224$	3079.01	0.918414
$225$	0	0
$226$	−2005.63	−0.590321
$227$	−210.751	−0.0616212	−0.0308106	−	0.999525i	$-0.509809\pi$
$227$	−210.751	−0.0616212	−0.0308106	+	0.999525i	$0.509809\pi$
$228$	0	0
$229$	1677.54	0.484083	0.242041	−	0.970266i	$-0.422183\pi$
$229$	1677.54	0.484083	0.242041	+	0.970266i	$0.422183\pi$
$230$	1612.84	0.462382
$231$	0	0
$232$	11000.5	3.11301
$233$	−2087.62	−0.586973	−0.293486	−	0.955963i	$-0.594816\pi$
$233$	−2087.62	−0.586973	−0.293486	+	0.955963i	$0.594816\pi$
$234$	0	0
$235$	3460.10	0.960477
$236$	4915.13	1.35571
$237$	0	0
$238$	−2289.55	−0.623569
$239$	−45.6018	−0.0123420	−0.00617100	−	0.999981i	$-0.501964\pi$
$239$	−45.6018	−0.0123420	−0.00617100	+	0.999981i	$0.501964\pi$
$240$	0	0
$241$	−5310.41	−1.41939	−0.709696	−	0.704508i	$-0.751168\pi$
$241$	−5310.41	−1.41939	−0.709696	+	0.704508i	$0.751168\pi$
$242$	1185.26	0.314841
$243$	0	0
$244$	−590.257	−0.154866
$245$	−647.943	−0.168962
$246$	0	0
$247$	−2077.13	−0.535079
$248$	−17697.0	−4.53128
$249$	0	0
$250$	−5269.34	−1.33305
$251$	1697.80	0.426950	0.213475	−	0.976949i	$-0.431522\pi$
$251$	1697.80	0.426950	0.213475	+	0.976949i	$0.431522\pi$
$252$	0	0
$253$	765.417	0.190203
$254$	−7965.35	−1.96768
$255$	0	0
$256$	−687.443	−0.167833
$257$	−3760.07	−0.912633	−0.456317	−	0.889818i	$-0.650832\pi$
$257$	−3760.07	−0.912633	−0.456317	+	0.889818i	$0.650832\pi$
$258$	0	0
$259$	2670.66	0.640721
$260$	20993.9	5.00765
$261$	0	0
$262$	9079.51	2.14097
$263$	4781.22	1.12100	0.560500	−	0.828155i	$-0.310609\pi$
$263$	4781.22	1.12100	0.560500	+	0.828155i	$0.310609\pi$
$264$	0	0
$265$	−4193.70	−0.972138
$266$	977.248	0.225259
$267$	0	0
$268$	−5558.77	−1.26700
$269$	5965.17	1.35205	0.676027	−	0.736877i	$-0.263700\pi$
$269$	5965.17	1.35205	0.676027	+	0.736877i	$0.263700\pi$
$270$	0	0
$271$	2958.83	0.663234	0.331617	−	0.943414i	$-0.392406\pi$
$271$	2958.83	0.663234	0.331617	+	0.943414i	$0.392406\pi$
$272$	−11097.1	−2.47376
$273$	0	0
$274$	7784.20	1.71628
$275$	1659.18	0.363826
$276$	0	0
$277$	−3147.15	−0.682650	−0.341325	−	0.939945i	$-0.610876\pi$
$277$	−3147.15	−0.682650	−0.341325	+	0.939945i	$0.610876\pi$
$278$	14617.7	3.15364
$279$	0	0
$280$	−5950.31	−1.27000
$281$	1073.25	0.227846	0.113923	−	0.993490i	$-0.463658\pi$
$281$	1073.25	0.227846	0.113923	+	0.993490i	$0.463658\pi$
$282$	0	0
$283$	4033.81	0.847298	0.423649	−	0.905827i	$-0.360749\pi$
$283$	4033.81	0.847298	0.423649	+	0.905827i	$0.360749\pi$
$284$	−16072.1	−3.35811
$285$	0	0
$286$	13924.3	2.87889
$287$	2414.04	0.496503
$288$	0	0
$289$	−1108.86	−0.225700
$290$	−11999.9	−2.42985
$291$	0	0
$292$	6624.52	1.32764
$293$	8699.49	1.73457	0.867286	−	0.497810i	$-0.165862\pi$
$293$	8699.49	1.73457	0.867286	+	0.497810i	$0.165862\pi$
$294$	0	0
$295$	−3230.01	−0.637485
$296$	24525.7	4.81597
$297$	0	0
$298$	13104.2	2.54733
$299$	−1814.71	−0.350995
$300$	0	0
$301$	2448.84	0.468932
$302$	6485.46	1.23575
$303$	0	0
$304$	4736.59	0.893625
$305$	387.890	0.0728215
$306$	0	0
$307$	9665.62	1.79689	0.898447	−	0.439082i	$-0.144696\pi$
$307$	9665.62	1.79689	0.898447	+	0.439082i	$0.144696\pi$
$308$	−4687.50	−0.867192
$309$	0	0
$310$	19304.7	3.53688
$311$	1660.16	0.302699	0.151349	−	0.988480i	$-0.451638\pi$
$311$	1660.16	0.302699	0.151349	+	0.988480i	$0.451638\pi$
$312$	0	0
$313$	3541.03	0.639460	0.319730	−	0.947509i	$-0.396408\pi$
$313$	3541.03	0.639460	0.319730	+	0.947509i	$0.396408\pi$
$314$	3610.76	0.648939
$315$	0	0
$316$	12705.4	2.26181
$317$	−3305.72	−0.585702	−0.292851	−	0.956158i	$-0.594604\pi$
$317$	−3305.72	−0.585702	−0.292851	+	0.956158i	$0.594604\pi$
$318$	0	0
$319$	−5694.85	−0.999531
$320$	−11811.2	−2.06333
$321$	0	0
$322$	853.787	0.147763
$323$	−1623.72	−0.279710
$324$	0	0
$325$	−3933.71	−0.671394
$326$	11853.6	2.01383
$327$	0	0
$328$	22169.1	3.73196
$329$	1831.66	0.306939
$330$	0	0
$331$	2945.62	0.489142	0.244571	−	0.969631i	$-0.421353\pi$
$331$	2945.62	0.489142	0.244571	+	0.969631i	$0.421353\pi$
$332$	−17418.1	−2.87934
$333$	0	0
$334$	−2424.98	−0.397272
$335$	3652.98	0.595772
$336$	0	0
$337$	−2200.66	−0.355720	−0.177860	−	0.984056i	$-0.556917\pi$
$337$	−2200.66	−0.355720	−0.177860	+	0.984056i	$0.556917\pi$
$338$	−21362.2	−3.43772
$339$	0	0
$340$	16411.3	2.61772
$341$	9161.54	1.45491
$342$	0	0
$343$	−343.000	−0.0539949
$344$	22488.6	3.52472
$345$	0	0
$346$	1919.55	0.298254
$347$	5503.40	0.851406	0.425703	−	0.904863i	$-0.360027\pi$
$347$	5503.40	0.851406	0.425703	+	0.904863i	$0.360027\pi$
$348$	0	0
$349$	−1729.03	−0.265195	−0.132597	−	0.991170i	$-0.542332\pi$
$349$	−1729.03	−0.265195	−0.132597	+	0.991170i	$0.542332\pi$
$350$	1850.73	0.282645
$351$	0	0
$352$	−14638.0	−2.21651
$353$	9670.43	1.45809	0.729044	−	0.684467i	$-0.239965\pi$
$353$	9670.43	1.45809	0.729044	+	0.684467i	$0.239965\pi$
$354$	0	0
$355$	10561.9	1.57906
$356$	18036.2	2.68515
$357$	0	0
$358$	17052.4	2.51745
$359$	−2493.27	−0.366545	−0.183272	−	0.983062i	$-0.558669\pi$
$359$	−2493.27	−0.366545	−0.183272	+	0.983062i	$0.558669\pi$
$360$	0	0
$361$	−6165.95	−0.898957
$362$	−13828.8	−2.00780
$363$	0	0
$364$	11113.5	1.60029
$365$	−4353.34	−0.624285
$366$	0	0
$367$	−2242.91	−0.319016	−0.159508	−	0.987197i	$-0.550991\pi$
$367$	−2242.91	−0.319016	−0.159508	+	0.987197i	$0.550991\pi$
$368$	4138.19	0.586190
$369$	0	0
$370$	−26753.8	−3.75909
$371$	−2220.01	−0.310666
$372$	0	0
$373$	−9982.67	−1.38575	−0.692873	−	0.721060i	$-0.743655\pi$
$373$	−9982.67	−1.38575	−0.692873	+	0.721060i	$0.743655\pi$
$374$	10884.8	1.50493
$375$	0	0
$376$	16820.9	2.30710
$377$	13501.8	1.84451
$378$	0	0
$379$	10402.8	1.40992	0.704958	−	0.709249i	$-0.250966\pi$
$379$	10402.8	1.40992	0.704958	+	0.709249i	$0.250966\pi$
$380$	−7004.82	−0.945630
$381$	0	0
$382$	530.533	0.0710587
$383$	6051.48	0.807352	0.403676	−	0.914902i	$-0.367732\pi$
$383$	6051.48	0.807352	0.403676	+	0.914902i	$0.367732\pi$
$384$	0	0
$385$	3080.42	0.407773
$386$	−9567.98	−1.26165
$387$	0	0
$388$	1187.11	0.155326
$389$	4673.87	0.609189	0.304594	−	0.952482i	$-0.401479\pi$
$389$	4673.87	0.609189	0.304594	+	0.952482i	$0.401479\pi$
$390$	0	0
$391$	−1418.59	−0.183481
$392$	−3149.90	−0.405852
$393$	0	0
$394$	−28323.0	−3.62156
$395$	−8349.40	−1.06355
$396$	0	0
$397$	13223.0	1.67164	0.835820	−	0.549004i	$-0.184993\pi$
$397$	13223.0	1.67164	0.835820	+	0.549004i	$0.184993\pi$
$398$	−17421.9	−2.19418
$399$	0	0
$400$	8970.25	1.12128
$401$	13044.0	1.62440	0.812200	−	0.583379i	$-0.198270\pi$
$401$	13044.0	1.62440	0.812200	+	0.583379i	$0.198270\pi$
$402$	0	0
$403$	−21720.9	−2.68485
$404$	7263.83	0.894528
$405$	0	0
$406$	−6352.33	−0.776505
$407$	−12696.7	−1.54632
$408$	0	0
$409$	−7511.39	−0.908103	−0.454052	−	0.890975i	$-0.650022\pi$
$409$	−7511.39	−0.908103	−0.454052	+	0.890975i	$0.650022\pi$
$410$	−24183.1	−2.91297
$411$	0	0
$412$	−18509.8	−2.21338
$413$	−1709.86	−0.203721
$414$	0	0
$415$	11446.4	1.35393
$416$	34705.1	4.09028
$417$	0	0
$418$	−4645.98	−0.543642
$419$	−4058.24	−0.473170	−0.236585	−	0.971611i	$-0.576028\pi$
$419$	−4058.24	−0.473170	−0.236585	+	0.971611i	$0.576028\pi$
$420$	0	0
$421$	1490.82	0.172585	0.0862923	−	0.996270i	$-0.472498\pi$
$421$	1490.82	0.172585	0.0862923	+	0.996270i	$0.472498\pi$
$422$	10041.7	1.15835
$423$	0	0
$424$	−20387.2	−2.33511
$425$	−3075.04	−0.350967
$426$	0	0
$427$	205.337	0.0232715
$428$	−35191.4	−3.97440
$429$	0	0
$430$	−24531.6	−2.75121
$431$	7097.92	0.793260	0.396630	−	0.917979i	$-0.370180\pi$
$431$	7097.92	0.793260	0.396630	+	0.917979i	$0.370180\pi$
$432$	0	0
$433$	−2693.96	−0.298991	−0.149496	−	0.988762i	$-0.547765\pi$
$433$	−2693.96	−0.298991	−0.149496	+	0.988762i	$0.547765\pi$
$434$	10219.3	1.13028
$435$	0	0
$436$	−39076.1	−4.29221
$437$	605.496	0.0662810
$438$	0	0
$439$	15775.1	1.71504	0.857521	−	0.514449i	$-0.172003\pi$
$439$	15775.1	1.71504	0.857521	+	0.514449i	$0.172003\pi$
$440$	28288.6	3.06502
$441$	0	0
$442$	−25806.7	−2.77715
$443$	−13364.7	−1.43336	−0.716679	−	0.697403i	$-0.754339\pi$
$443$	−13364.7	−1.43336	−0.716679	+	0.697403i	$0.754339\pi$
$444$	0	0
$445$	−11852.6	−1.26262
$446$	2419.61	0.256887
$447$	0	0
$448$	−6252.45	−0.659376
$449$	−6338.41	−0.666209	−0.333105	−	0.942890i	$-0.608096\pi$
$449$	−6338.41	−0.666209	−0.333105	+	0.942890i	$0.608096\pi$
$450$	0	0
$451$	−11476.7	−1.19826
$452$	7610.27	0.791940
$453$	0	0
$454$	1117.62	0.115534
$455$	−7303.30	−0.752492
$456$	0	0
$457$	1238.17	0.126738	0.0633691	−	0.997990i	$-0.479815\pi$
$457$	1238.17	0.126738	0.0633691	+	0.997990i	$0.479815\pi$
$458$	−8896.03	−0.907608
$459$	0	0
$460$	−6119.86	−0.620304
$461$	−9408.36	−0.950522	−0.475261	−	0.879845i	$-0.657646\pi$
$461$	−9408.36	−0.950522	−0.475261	+	0.879845i	$0.657646\pi$
$462$	0	0
$463$	3820.66	0.383501	0.191750	−	0.981444i	$-0.438584\pi$
$463$	3820.66	0.383501	0.191750	+	0.981444i	$0.438584\pi$
$464$	−30788.9	−3.08047
$465$	0	0
$466$	11070.7	1.10052
$467$	7625.72	0.755623	0.377812	−	0.925882i	$-0.376677\pi$
$467$	7625.72	0.755623	0.377812	+	0.925882i	$0.376677\pi$
$468$	0	0
$469$	1933.77	0.190390
$470$	−18349.0	−1.80080
$471$	0	0
$472$	−15702.3	−1.53126
$473$	−11642.1	−1.13172
$474$	0	0
$475$	1312.52	0.126784
$476$	8687.58	0.836543
$477$	0	0
$478$	241.828	0.0231400
$479$	−10981.9	−1.04755	−0.523774	−	0.851857i	$-0.675476\pi$
$479$	−10981.9	−1.04755	−0.523774	+	0.851857i	$0.675476\pi$
$480$	0	0
$481$	30102.4	2.85354
$482$	28161.2	2.66122
$483$	0	0
$484$	−4497.42	−0.422373
$485$	−780.117	−0.0730377
$486$	0	0
$487$	17967.8	1.67186	0.835932	−	0.548833i	$-0.184928\pi$
$487$	17967.8	1.67186	0.835932	+	0.548833i	$0.184928\pi$
$488$	1885.68	0.174920
$489$	0	0
$490$	3436.06	0.316786
$491$	−472.865	−0.0434625	−0.0217313	−	0.999764i	$-0.506918\pi$
$491$	−472.865	−0.0434625	−0.0217313	+	0.999764i	$0.506918\pi$
$492$	0	0
$493$	10554.5	0.964205
$494$	11015.1	1.00322
$495$	0	0
$496$	49531.4	4.48392
$497$	5591.10	0.504618
$498$	0	0
$499$	10839.6	0.972440	0.486220	−	0.873836i	$-0.338375\pi$
$499$	10839.6	0.972440	0.486220	+	0.873836i	$0.338375\pi$
$500$	19994.2	1.78834
$501$	0	0
$502$	−9003.50	−0.800489
$503$	3968.56	0.351788	0.175894	−	0.984409i	$-0.443718\pi$
$503$	3968.56	0.351788	0.175894	+	0.984409i	$0.443718\pi$
$504$	0	0
$505$	−4773.47	−0.420627
$506$	−4059.03	−0.356612
$507$	0	0
$508$	30224.1	2.63972
$509$	−12054.8	−1.04975	−0.524873	−	0.851181i	$-0.675887\pi$
$509$	−12054.8	−1.04975	−0.524873	+	0.851181i	$0.675887\pi$
$510$	0	0
$511$	−2304.52	−0.199502
$512$	13388.2	1.15563
$513$	0	0
$514$	19939.7	1.71110
$515$	12163.8	1.04078
$516$	0	0
$517$	−8708.00	−0.740768
$518$	−14162.6	−1.20129
$519$	0	0
$520$	−67069.0	−5.65610
$521$	17218.9	1.44793	0.723965	−	0.689837i	$-0.242318\pi$
$521$	17218.9	1.44793	0.723965	+	0.689837i	$0.242318\pi$
$522$	0	0
$523$	10560.4	0.882936	0.441468	−	0.897277i	$-0.354458\pi$
$523$	10560.4	0.882936	0.441468	+	0.897277i	$0.354458\pi$
$524$	−34451.8	−2.87220
$525$	0	0
$526$	−25355.0	−2.10176
$527$	−16979.5	−1.40349
$528$	0	0
$529$	529.000	0.0434783
$530$	22239.3	1.82266
$531$	0	0
$532$	−3708.12	−0.302194
$533$	27209.9	2.21124
$534$	0	0
$535$	23126.2	1.86885
$536$	17758.5	1.43107
$537$	0	0
$538$	−31633.4	−2.53497
$539$	1630.67	0.130312
$540$	0	0
$541$	2004.42	0.159291	0.0796457	−	0.996823i	$-0.474621\pi$
$541$	2004.42	0.159291	0.0796457	+	0.996823i	$0.474621\pi$
$542$	−15690.8	−1.24350
$543$	0	0
$544$	27129.4	2.13817
$545$	25679.1	2.01829
$546$	0	0
$547$	−17235.2	−1.34721	−0.673606	−	0.739090i	$-0.735256\pi$
$547$	−17235.2	−1.34721	−0.673606	+	0.739090i	$0.735256\pi$
$548$	−29536.8	−2.30246
$549$	0	0
$550$	−8798.65	−0.682138
$551$	−4505.00	−0.348311
$552$	0	0
$553$	−4419.90	−0.339879
$554$	16689.4	1.27990
$555$	0	0
$556$	−55466.2	−4.23074
$557$	7492.05	0.569926	0.284963	−	0.958539i	$-0.408019\pi$
$557$	7492.05	0.569926	0.284963	+	0.958539i	$0.408019\pi$
$558$	0	0
$559$	27602.1	2.08845
$560$	16654.1	1.25672
$561$	0	0
$562$	−5691.46	−0.427188
$563$	−10238.7	−0.766446	−0.383223	−	0.923656i	$-0.625186\pi$
$563$	−10238.7	−0.766446	−0.383223	+	0.923656i	$0.625186\pi$
$564$	0	0
$565$	−5001.13	−0.372388
$566$	−21391.4	−1.58860
$567$	0	0
$568$	51345.2	3.79295
$569$	10594.1	0.780538	0.390269	−	0.920701i	$-0.372382\pi$
$569$	10594.1	0.780538	0.390269	+	0.920701i	$0.372382\pi$
$570$	0	0
$571$	−20152.2	−1.47696	−0.738478	−	0.674277i	$-0.764455\pi$
$571$	−20152.2	−1.47696	−0.738478	+	0.674277i	$0.764455\pi$
$572$	−52835.2	−3.86215
$573$	0	0
$574$	−12801.7	−0.930895
$575$	1146.70	0.0831664
$576$	0	0
$577$	3057.90	0.220628	0.110314	−	0.993897i	$-0.464814\pi$
$577$	3057.90	0.220628	0.110314	+	0.993897i	$0.464814\pi$
$578$	5880.32	0.423165
$579$	0	0
$580$	45532.9	3.25974
$581$	6059.34	0.432674
$582$	0	0
$583$	10554.2	0.749762
$584$	−21163.2	−1.49956
$585$	0	0
$586$	−46133.6	−3.25215
$587$	10847.1	0.762707	0.381354	−	0.924429i	$-0.375458\pi$
$587$	10847.1	0.762707	0.381354	+	0.924429i	$0.375458\pi$
$588$	0	0
$589$	7247.38	0.507000
$590$	17128.8	1.19522
$591$	0	0
$592$	−68644.1	−4.76564
$593$	25889.2	1.79282	0.896411	−	0.443224i	$-0.146166\pi$
$593$	25889.2	1.79282	0.896411	+	0.443224i	$0.146166\pi$
$594$	0	0
$595$	−5709.09	−0.393361
$596$	−49723.2	−3.41735
$597$	0	0
$598$	9623.47	0.658082
$599$	12917.8	0.881144	0.440572	−	0.897717i	$-0.354776\pi$
$599$	12917.8	0.881144	0.440572	+	0.897717i	$0.354776\pi$
$600$	0	0
$601$	24952.7	1.69358	0.846791	−	0.531925i	$-0.178531\pi$
$601$	24952.7	1.69358	0.846791	+	0.531925i	$0.178531\pi$
$602$	−12986.2	−0.879202
$603$	0	0
$604$	−24608.8	−1.65781
$605$	2955.51	0.198609
$606$	0	0
$607$	−3113.16	−0.208170	−0.104085	−	0.994568i	$-0.533191\pi$
$607$	−3113.16	−0.208170	−0.104085	+	0.994568i	$0.533191\pi$
$608$	−11579.7	−0.772396
$609$	0	0
$610$	−2056.99	−0.136533
$611$	20645.6	1.36699
$612$	0	0
$613$	−1088.16	−0.0716971	−0.0358486	−	0.999357i	$-0.511413\pi$
$613$	−1088.16	−0.0716971	−0.0358486	+	0.999357i	$0.511413\pi$
$614$	−51257.1	−3.36900
$615$	0	0
$616$	14975.1	0.979486
$617$	−21391.6	−1.39577	−0.697886	−	0.716209i	$-0.745876\pi$
$617$	−21391.6	−1.39577	−0.697886	+	0.716209i	$0.745876\pi$
$618$	0	0
$619$	−29839.1	−1.93754	−0.968768	−	0.247971i	$-0.920236\pi$
$619$	−29839.1	−1.93754	−0.968768	+	0.247971i	$0.920236\pi$
$620$	−73250.7	−4.74487
$621$	0	0
$622$	−8803.89	−0.567530
$623$	−6274.36	−0.403494
$624$	0	0
$625$	−19371.4	−1.23977
$626$	−18778.2	−1.19893
$627$	0	0
$628$	−13700.8	−0.870578
$629$	23531.5	1.49167
$630$	0	0
$631$	−8469.09	−0.534309	−0.267155	−	0.963654i	$-0.586083\pi$
$631$	−8469.09	−0.534309	−0.267155	+	0.963654i	$0.586083\pi$
$632$	−40589.6	−2.55470
$633$	0	0
$634$	17530.3	1.09813
$635$	−19862.0	−1.24126
$636$	0	0
$637$	−3866.13	−0.240474
$638$	30199.9	1.87402
$639$	0	0
$640$	16103.8	0.994626
$641$	−22742.0	−1.40134	−0.700668	−	0.713488i	$-0.747114\pi$
$641$	−22742.0	−1.40134	−0.700668	+	0.713488i	$0.747114\pi$
$642$	0	0
$643$	−4464.21	−0.273797	−0.136898	−	0.990585i	$-0.543713\pi$
$643$	−4464.21	−0.273797	−0.136898	+	0.990585i	$0.543713\pi$
$644$	−3239.65	−0.198230
$645$	0	0
$646$	8610.63	0.524428
$647$	2439.18	0.148213	0.0741066	−	0.997250i	$-0.476390\pi$
$647$	2439.18	0.148213	0.0741066	+	0.997250i	$0.476390\pi$
$648$	0	0
$649$	8128.92	0.491661
$650$	20860.6	1.25880
$651$	0	0
$652$	−44977.9	−2.70164
$653$	−11479.7	−0.687959	−0.343979	−	0.938977i	$-0.611775\pi$
$653$	−11479.7	−0.687959	−0.343979	+	0.938977i	$0.611775\pi$
$654$	0	0
$655$	22640.2	1.35057
$656$	−62048.2	−3.69295
$657$	0	0
$658$	−9713.36	−0.575480
$659$	−1729.36	−0.102225	−0.0511125	−	0.998693i	$-0.516277\pi$
$659$	−1729.36	−0.102225	−0.0511125	+	0.998693i	$0.516277\pi$
$660$	0	0
$661$	6835.28	0.402211	0.201106	−	0.979570i	$-0.435547\pi$
$661$	6835.28	0.402211	0.201106	+	0.979570i	$0.435547\pi$
$662$	−15620.7	−0.917094
$663$	0	0
$664$	55645.3	3.25219
$665$	2436.81	0.142099
$666$	0	0
$667$	−3935.86	−0.228481
$668$	9201.46	0.532957
$669$	0	0
$670$	−19371.8	−1.11701
$671$	−976.200	−0.0561636
$672$	0	0
$673$	−22296.1	−1.27705	−0.638523	−	0.769603i	$-0.720454\pi$
$673$	−22296.1	−1.27705	−0.638523	+	0.769603i	$0.720454\pi$
$674$	11670.2	0.666941
$675$	0	0
$676$	81057.9	4.61185
$677$	18268.1	1.03708	0.518538	−	0.855055i	$-0.326477\pi$
$677$	18268.1	1.03708	0.518538	+	0.855055i	$0.326477\pi$
$678$	0	0
$679$	−412.968	−0.0233406
$680$	−52428.8	−2.95669
$681$	0	0
$682$	−48583.9	−2.72782
$683$	31781.6	1.78051	0.890256	−	0.455460i	$-0.150525\pi$
$683$	31781.6	1.78051	0.890256	+	0.455460i	$0.150525\pi$
$684$	0	0
$685$	19410.3	1.08267
$686$	1818.94	0.101235
$687$	0	0
$688$	−62942.5	−3.48788
$689$	−25022.8	−1.38359
$690$	0	0
$691$	−23032.7	−1.26803	−0.634014	−	0.773322i	$-0.718594\pi$
$691$	−23032.7	−1.26803	−0.634014	+	0.773322i	$0.718594\pi$
$692$	−7283.65	−0.400120
$693$	0	0
$694$	−29184.7	−1.59630
$695$	36449.9	1.98939
$696$	0	0
$697$	21270.4	1.15591
$698$	9169.11	0.497215
$699$	0	0
$700$	−7022.52	−0.379180
$701$	−29844.6	−1.60801	−0.804006	−	0.594621i	$-0.797302\pi$
$701$	−29844.6	−1.60801	−0.804006	+	0.594621i	$0.797302\pi$
$702$	0	0
$703$	−10043.9	−0.538854
$704$	29725.0	1.59134
$705$	0	0
$706$	−51282.6	−2.73377
$707$	−2526.92	−0.134419
$708$	0	0
$709$	5462.65	0.289357	0.144679	−	0.989479i	$-0.453785\pi$
$709$	5462.65	0.289357	0.144679	+	0.989479i	$0.453785\pi$
$710$	−56009.8	−2.96058
$711$	0	0
$712$	−57619.8	−3.03286
$713$	6331.78	0.332576
$714$	0	0
$715$	34721.0	1.81607
$716$	−64704.6	−3.37727
$717$	0	0
$718$	13221.9	0.687236
$719$	−30493.4	−1.58166	−0.790829	−	0.612038i	$-0.790350\pi$
$719$	−30493.4	−1.58166	−0.790829	+	0.612038i	$0.790350\pi$
$720$	0	0
$721$	6439.12	0.332601
$722$	32698.2	1.68546
$723$	0	0
$724$	52472.6	2.69355
$725$	−8531.66	−0.437045
$726$	0	0
$727$	19889.3	1.01465	0.507326	−	0.861754i	$-0.330634\pi$
$727$	19889.3	1.01465	0.507326	+	0.861754i	$0.330634\pi$
$728$	−35504.1	−1.80752
$729$	0	0
$730$	23085.9	1.17047
$731$	21576.9	1.09173
$732$	0	0
$733$	5240.27	0.264057	0.132029	−	0.991246i	$-0.457851\pi$
$733$	5240.27	0.264057	0.132029	+	0.991246i	$0.457851\pi$
$734$	11894.2	0.598124
$735$	0	0
$736$	−10116.7	−0.506668
$737$	−9193.41	−0.459489
$738$	0	0
$739$	18395.1	0.915664	0.457832	−	0.889039i	$-0.348626\pi$
$739$	18395.1	0.915664	0.457832	+	0.889039i	$0.348626\pi$
$740$	101516.	5.04298
$741$	0	0
$742$	11772.7	0.582468
$743$	4710.74	0.232598	0.116299	−	0.993214i	$-0.462897\pi$
$743$	4710.74	0.232598	0.116299	+	0.993214i	$0.462897\pi$
$744$	0	0
$745$	32675.9	1.60692
$746$	52938.4	2.59814
$747$	0	0
$748$	−41302.0	−2.01892
$749$	12242.3	0.597227
$750$	0	0
$751$	−19802.3	−0.962178	−0.481089	−	0.876672i	$-0.659759\pi$
$751$	−19802.3	−0.962178	−0.481089	+	0.876672i	$0.659759\pi$
$752$	−47079.3	−2.28299
$753$	0	0
$754$	−71600.4	−3.45827
$755$	16171.8	0.779539
$756$	0	0
$757$	13257.1	0.636508	0.318254	−	0.948005i	$-0.396903\pi$
$757$	13257.1	0.636508	0.318254	+	0.948005i	$0.396903\pi$
$758$	−55166.5	−2.64345
$759$	0	0
$760$	22378.2	1.06808
$761$	−825.676	−0.0393308	−0.0196654	−	0.999807i	$-0.506260\pi$
$761$	−825.676	−0.0393308	−0.0196654	+	0.999807i	$0.506260\pi$
$762$	0	0
$763$	13593.7	0.644985
$764$	−2013.08	−0.0953282
$765$	0	0
$766$	−32091.1	−1.51371
$767$	−19272.7	−0.907297
$768$	0	0
$769$	−11060.2	−0.518651	−0.259325	−	0.965790i	$-0.583500\pi$
$769$	−11060.2	−0.518651	−0.259325	+	0.965790i	$0.583500\pi$
$770$	−16335.5	−0.764534
$771$	0	0
$772$	36305.2	1.69256
$773$	13681.2	0.636581	0.318290	−	0.947993i	$-0.396891\pi$
$773$	13681.2	0.636581	0.318290	+	0.947993i	$0.396891\pi$
$774$	0	0
$775$	13725.2	0.636161
$776$	−3792.45	−0.175439
$777$	0	0
$778$	−24785.6	−1.14217
$779$	−9078.83	−0.417565
$780$	0	0
$781$	−26580.9	−1.21785
$782$	7522.80	0.344009
$783$	0	0
$784$	8816.14	0.401610
$785$	9003.59	0.409365
$786$	0	0
$787$	−32029.0	−1.45071	−0.725357	−	0.688373i	$-0.758325\pi$
$787$	−32029.0	−1.45071	−0.725357	+	0.688373i	$0.758325\pi$
$788$	107470.	4.85847
$789$	0	0
$790$	44277.1	1.99406
$791$	−2647.44	−0.119004
$792$	0	0
$793$	2314.45	0.103643
$794$	−70121.7	−3.13416
$795$	0	0
$796$	66106.7	2.94358
$797$	−31893.7	−1.41748	−0.708741	−	0.705469i	$-0.750737\pi$
$797$	−31893.7	−1.41748	−0.708741	+	0.705469i	$0.750737\pi$
$798$	0	0
$799$	16139.0	0.714588
$800$	−21929.8	−0.969169
$801$	0	0
$802$	−69172.5	−3.04559
$803$	10956.0	0.481481
$804$	0	0
$805$	2128.96	0.0932122
$806$	115187.	5.03384
$807$	0	0
$808$	−23205.7	−1.01036
$809$	7163.61	0.311322	0.155661	−	0.987811i	$-0.450249\pi$
$809$	7163.61	0.311322	0.155661	+	0.987811i	$0.450249\pi$
$810$	0	0
$811$	−18347.8	−0.794423	−0.397212	−	0.917727i	$-0.630022\pi$
$811$	−18347.8	−0.794423	−0.397212	+	0.917727i	$0.630022\pi$
$812$	24103.6	1.04171
$813$	0	0
$814$	67331.0	2.89920
$815$	29557.5	1.27037
$816$	0	0
$817$	−9209.69	−0.394377
$818$	39833.1	1.70260
$819$	0	0
$820$	91761.5	3.90787
$821$	26934.9	1.14499	0.572494	−	0.819909i	$-0.305976\pi$
$821$	26934.9	1.14499	0.572494	+	0.819909i	$0.305976\pi$
$822$	0	0
$823$	−3763.90	−0.159418	−0.0797092	−	0.996818i	$-0.525399\pi$
$823$	−3763.90	−0.159418	−0.0797092	+	0.996818i	$0.525399\pi$
$824$	59132.8	2.49999
$825$	0	0
$826$	9067.43	0.381957
$827$	32630.4	1.37203	0.686015	−	0.727587i	$-0.259358\pi$
$827$	32630.4	1.37203	0.686015	+	0.727587i	$0.259358\pi$
$828$	0	0
$829$	10237.6	0.428911	0.214455	−	0.976734i	$-0.431202\pi$
$829$	10237.6	0.428911	0.214455	+	0.976734i	$0.431202\pi$
$830$	−60700.5	−2.53849
$831$	0	0
$832$	−70474.6	−2.93662
$833$	−3022.21	−0.125706
$834$	0	0
$835$	−6046.79	−0.250608
$836$	17629.0	0.729318
$837$	0	0
$838$	21521.0	0.887147
$839$	−39454.8	−1.62352	−0.811758	−	0.583995i	$-0.801489\pi$
$839$	−39454.8	−1.62352	−0.811758	+	0.583995i	$0.801489\pi$
$840$	0	0
$841$	4894.51	0.200685
$842$	−7905.86	−0.323579
$843$	0	0
$844$	−38102.7	−1.55397
$845$	−53267.6	−2.16859
$846$	0	0
$847$	1564.55	0.0634693
$848$	57060.9	2.31071
$849$	0	0
$850$	16307.0	0.658030
$851$	−8775.03	−0.353471
$852$	0	0
$853$	18731.3	0.751874	0.375937	−	0.926645i	$-0.377321\pi$
$853$	18731.3	0.751874	0.375937	+	0.926645i	$0.377321\pi$
$854$	−1088.90	−0.0436318
$855$	0	0
$856$	112426.	4.48905
$857$	−16359.9	−0.652091	−0.326046	−	0.945354i	$-0.605716\pi$
$857$	−16359.9	−0.652091	−0.326046	+	0.945354i	$0.605716\pi$
$858$	0	0
$859$	38453.0	1.52736	0.763678	−	0.645597i	$-0.223391\pi$
$859$	38453.0	1.52736	0.763678	+	0.645597i	$0.223391\pi$
$860$	93084.1	3.69086
$861$	0	0
$862$	−37640.5	−1.48728
$863$	−19604.0	−0.773264	−0.386632	−	0.922234i	$-0.626362\pi$
$863$	−19604.0	−0.773264	−0.386632	+	0.922234i	$0.626362\pi$
$864$	0	0
$865$	4786.49	0.188145
$866$	14286.1	0.560580
$867$	0	0
$868$	−38776.5	−1.51631
$869$	21012.8	0.820267
$870$	0	0
$871$	21796.5	0.847928
$872$	124836.	4.84802
$873$	0	0
$874$	−3210.96	−0.124270
$875$	−6955.53	−0.268731
$876$	0	0
$877$	−39580.7	−1.52400	−0.761998	−	0.647580i	$-0.775781\pi$
$877$	−39580.7	−1.52400	−0.761998	+	0.647580i	$0.775781\pi$
$878$	−83655.7	−3.21554
$879$	0	0
$880$	−79176.1	−3.03298
$881$	−35741.9	−1.36683	−0.683413	−	0.730032i	$-0.739505\pi$
$881$	−35741.9	−1.36683	−0.683413	+	0.730032i	$0.739505\pi$
$882$	0	0
$883$	−11289.5	−0.430262	−0.215131	−	0.976585i	$-0.569018\pi$
$883$	−11289.5	−0.430262	−0.215131	+	0.976585i	$0.569018\pi$
$884$	97922.2	3.72566
$885$	0	0
$886$	70873.5	2.68741
$887$	4125.49	0.156167	0.0780837	−	0.996947i	$-0.475120\pi$
$887$	4125.49	0.156167	0.0780837	+	0.996947i	$0.475120\pi$
$888$	0	0
$889$	−10514.3	−0.396667
$890$	62854.4	2.36729
$891$	0	0
$892$	−9181.09	−0.344625
$893$	−6888.60	−0.258139
$894$	0	0
$895$	42521.0	1.58807
$896$	8524.85	0.317852
$897$	0	0
$898$	33612.7	1.24908
$899$	−47109.6	−1.74771
$900$	0	0
$901$	−19560.7	−0.723264
$902$	60861.3	2.24663
$903$	0	0
$904$	−24312.4	−0.894490
$905$	−34482.7	−1.26657
$906$	0	0
$907$	19400.9	0.710250	0.355125	−	0.934819i	$-0.384438\pi$
$907$	19400.9	0.710250	0.355125	+	0.934819i	$0.384438\pi$
$908$	−4240.74	−0.154993
$909$	0	0
$910$	38729.6	1.41085
$911$	−31001.4	−1.12747	−0.563734	−	0.825956i	$-0.690636\pi$
$911$	−31001.4	−1.12747	−0.563734	+	0.825956i	$0.690636\pi$
$912$	0	0
$913$	−28807.0	−1.04422
$914$	−6566.07	−0.237622
$915$	0	0
$916$	33755.6	1.21759
$917$	11985.0	0.431602
$918$	0	0
$919$	1160.59	0.0416587	0.0208294	−	0.999783i	$-0.493369\pi$
$919$	1160.59	0.0416587	0.0208294	+	0.999783i	$0.493369\pi$
$920$	19551.0	0.700628
$921$	0	0
$922$	49892.7	1.78214
$923$	63020.2	2.24738
$924$	0	0
$925$	−19021.4	−0.676130
$926$	−20261.0	−0.719027
$927$	0	0
$928$	75270.4	2.66258
$929$	41683.2	1.47210	0.736051	−	0.676926i	$-0.236689\pi$
$929$	41683.2	1.47210	0.736051	+	0.676926i	$0.236689\pi$
$930$	0	0
$931$	1289.97	0.0454103
$932$	−42007.3	−1.47639
$933$	0	0
$934$	−40439.4	−1.41672
$935$	27141.9	0.949341
$936$	0	0
$937$	−38732.1	−1.35040	−0.675198	−	0.737637i	$-0.735942\pi$
$937$	−38732.1	−1.35040	−0.675198	+	0.737637i	$0.735942\pi$
$938$	−10254.8	−0.356963
$939$	0	0
$940$	69624.4	2.41585
$941$	18080.0	0.626345	0.313173	−	0.949696i	$-0.398608\pi$
$941$	18080.0	0.626345	0.313173	+	0.949696i	$0.398608\pi$
$942$	0	0
$943$	−7931.85	−0.273910
$944$	43948.6	1.51526
$945$	0	0
$946$	61738.5	2.12187
$947$	6401.43	0.219661	0.109830	−	0.993950i	$-0.464969\pi$
$947$	6401.43	0.219661	0.109830	+	0.993950i	$0.464969\pi$
$948$	0	0
$949$	−25975.4	−0.888511
$950$	−6960.31	−0.237708
$951$	0	0
$952$	−27754.1	−0.944869
$953$	30907.1	1.05055	0.525277	−	0.850931i	$-0.323962\pi$
$953$	30907.1	1.05055	0.525277	+	0.850931i	$0.323962\pi$
$954$	0	0
$955$	1322.91	0.0448255
$956$	−917.603	−0.0310433
$957$	0	0
$958$	58237.2	1.96405
$959$	10275.2	0.345988
$960$	0	0
$961$	45996.2	1.54396
$962$	−159634.	−5.35010
$963$	0	0
$964$	−106856.	−3.57014
$965$	−23858.2	−0.795878
$966$	0	0
$967$	25704.4	0.854805	0.427402	−	0.904061i	$-0.359429\pi$
$967$	25704.4	0.854805	0.427402	+	0.904061i	$0.359429\pi$
$968$	14367.8	0.477066
$969$	0	0
$970$	4136.98	0.136939
$971$	31129.1	1.02881	0.514407	−	0.857546i	$-0.328012\pi$
$971$	31129.1	1.02881	0.514407	+	0.857546i	$0.328012\pi$
$972$	0	0
$973$	19295.4	0.635747
$974$	−95283.6	−3.13458
$975$	0	0
$976$	−5277.77	−0.173092
$977$	−28515.3	−0.933761	−0.466881	−	0.884320i	$-0.654622\pi$
$977$	−28515.3	−0.933761	−0.466881	+	0.884320i	$0.654622\pi$
$978$	0	0
$979$	29829.2	0.973796
$980$	−13038.0	−0.424982
$981$	0	0
$982$	2507.61	0.0814880
$983$	505.010	0.0163859	0.00819293	−	0.999966i	$-0.497392\pi$
$983$	505.010	0.0163859	0.00819293	+	0.999966i	$0.497392\pi$
$984$	0	0
$985$	−70624.7	−2.28456
$986$	−55971.0	−1.80779
$987$	0	0
$988$	−41796.1	−1.34586
$989$	−8046.18	−0.258699
$990$	0	0
$991$	6105.83	0.195720	0.0978598	−	0.995200i	$-0.468800\pi$
$991$	6105.83	0.195720	0.0978598	+	0.995200i	$0.468800\pi$
$992$	−121091.	−3.87563
$993$	0	0
$994$	−29649.8	−0.946110
$995$	−43442.4	−1.38414
$996$	0	0
$997$	−26216.8	−0.832794	−0.416397	−	0.909183i	$-0.636707\pi$
$997$	−26216.8	−0.832794	−0.416397	+	0.909183i	$0.636707\pi$
$998$	−57482.7	−1.82323
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1449.4.a.n.1.1		9
3.2	odd	2		161.4.a.c.1.9	✓	9
21.20	even	2		1127.4.a.f.1.9		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
161.4.a.c.1.9	✓	9	3.2	odd	2
1127.4.a.f.1.9		9	21.20	even	2
1449.4.a.n.1.1		9	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1449 = 3^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1449.a (trivial)

\(f(q)\)	\(=\)	\(q-5.30303 q^{2} +20.1221 q^{4} -13.2233 q^{5} -7.00000 q^{7} -64.2837 q^{8} +O(q^{10})\)	\(q-5.30303 q^{2} +20.1221 q^{4} -13.2233 q^{5} -7.00000 q^{7} -64.2837 q^{8} +70.1237 q^{10} +33.2790 q^{11} -78.9006 q^{13} +37.1212 q^{14} +179.921 q^{16} -61.6777 q^{17} +26.3259 q^{19} -266.081 q^{20} -176.479 q^{22} +23.0000 q^{23} +49.8565 q^{25} +418.412 q^{26} -140.855 q^{28} -171.124 q^{29} +275.295 q^{31} -439.858 q^{32} +327.078 q^{34} +92.5633 q^{35} -381.523 q^{37} -139.607 q^{38} +850.044 q^{40} -344.863 q^{41} -349.834 q^{43} +669.643 q^{44} -121.970 q^{46} -261.666 q^{47} +49.0000 q^{49} -264.390 q^{50} -1587.64 q^{52} +317.144 q^{53} -440.060 q^{55} +449.986 q^{56} +907.476 q^{58} +244.266 q^{59} -29.3338 q^{61} -1459.89 q^{62} +893.207 q^{64} +1043.33 q^{65} -276.252 q^{67} -1241.08 q^{68} -490.866 q^{70} -798.729 q^{71} +329.217 q^{73} +2023.23 q^{74} +529.732 q^{76} -232.953 q^{77} +631.414 q^{79} -2379.16 q^{80} +1828.82 q^{82} -865.620 q^{83} +815.585 q^{85} +1855.18 q^{86} -2139.30 q^{88} +896.337 q^{89} +552.304 q^{91} +462.808 q^{92} +1387.62 q^{94} -348.116 q^{95} +58.9955 q^{97} -259.848 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 48 q^{4} + 4 q^{5} - 63 q^{7} - 66 q^{8}+O(q^{10}) \) 9 * q + 48 * q^4 + 4 * q^5 - 63 * q^7 - 66 * q^8	\( 9 q + 48 q^{4} + 4 q^{5} - 63 q^{7} - 66 q^{8} + 50 q^{10} + 8 q^{11} + 25 q^{13} + 180 q^{16} + 28 q^{17} + 254 q^{19} - 302 q^{20} - 122 q^{22} + 207 q^{23} + 295 q^{25} + 6 q^{26} - 336 q^{28} - 25 q^{29} + 1171 q^{31} - 1018 q^{32} + 8 q^{34} - 28 q^{35} + 70 q^{37} - 282 q^{38} - 54 q^{40} - 221 q^{41} - 42 q^{43} + 1214 q^{44} - 159 q^{47} + 441 q^{49} + 1680 q^{50} - 664 q^{52} + 774 q^{53} + 1498 q^{55} + 462 q^{56} + 842 q^{58} - 1080 q^{59} + 686 q^{61} + 2078 q^{62} - 1260 q^{64} + 1656 q^{65} - 370 q^{67} + 1936 q^{68} - 350 q^{70} - 1035 q^{71} + 1979 q^{73} + 5494 q^{74} + 206 q^{76} - 56 q^{77} + 2336 q^{79} + 242 q^{80} + 1642 q^{82} - 130 q^{83} - 272 q^{85} + 1906 q^{86} - 3742 q^{88} + 1328 q^{89} - 175 q^{91} + 1104 q^{92} + 6078 q^{94} + 484 q^{95} - 104 q^{97}+O(q^{100}) \) 9 * q + 48 * q^4 + 4 * q^5 - 63 * q^7 - 66 * q^8 + 50 * q^10 + 8 * q^11 + 25 * q^13 + 180 * q^16 + 28 * q^17 + 254 * q^19 - 302 * q^20 - 122 * q^22 + 207 * q^23 + 295 * q^25 + 6 * q^26 - 336 * q^28 - 25 * q^29 + 1171 * q^31 - 1018 * q^32 + 8 * q^34 - 28 * q^35 + 70 * q^37 - 282 * q^38 - 54 * q^40 - 221 * q^41 - 42 * q^43 + 1214 * q^44 - 159 * q^47 + 441 * q^49 + 1680 * q^50 - 664 * q^52 + 774 * q^53 + 1498 * q^55 + 462 * q^56 + 842 * q^58 - 1080 * q^59 + 686 * q^61 + 2078 * q^62 - 1260 * q^64 + 1656 * q^65 - 370 * q^67 + 1936 * q^68 - 350 * q^70 - 1035 * q^71 + 1979 * q^73 + 5494 * q^74 + 206 * q^76 - 56 * q^77 + 2336 * q^79 + 242 * q^80 + 1642 * q^82 - 130 * q^83 - 272 * q^85 + 1906 * q^86 - 3742 * q^88 + 1328 * q^89 - 175 * q^91 + 1104 * q^92 + 6078 * q^94 + 484 * q^95 - 104 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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